A008955 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(1 + 3\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	52	58	16													126
3	1456	2560	1492	288												5796
4	71344	172032	155028	61856	9216											469476
5	5422144	16641632	20383728	12452456	3793216	460800										59153976
6	591013696	2204332256	3420023856	2824946120	1310037376	323338752	33177600									10706869656
7	87470027008	384160516096	722188091776	753254363648	470730251408	176237798144	36597482496	3251404800								2633889935376
8	16881715212544	85339143063552	188554847773056	237818167931392	187267497068688	94267367222016	29621752284160	5311998885888	416179814400							845478669255696
9	4119138511860736	23557588771938816	59832324032921088	88573518314494720	84219812489645376	53338572127299360	22499021047301632	6094851598190592	962091694227456	67421129932800						343264339717812576
10	1239860692070081536	7914661922725730816	22721047288297010688	38627093819247128320	43064867222282202176	32898872708246182560	17439919760697663232	6334354540515694592	1508559919600582656	212712098955264000	13484225986560000					171975434198624100576
11	451309291913509679104	3180983227353125748736	10185809398239738747904	19558755593973831294976	25023368373168526645504	22396887533593903378432	14309657988289525595968	6526165634836547333632	2082029609539410178048	442498704563057098752	56384586206281728000	3263182688747520000				104217113124366204949056
12	195416923398549691052032	1507342813514994236784640	5326578638915507093478400	11402454278883713710120960	16468040116646435450456320	16904582393518695836766208	12646385518604408756042560	6947011220511608367101440	2781054523763490239180800	791226466623153855037440	151854152741480432664576	17651718931636813824000	939796614359285760000			75140538562668033768269376
13	99271797086463243054432256	831781069374326867862183936	3215383819537145655520235520	7592830353626367962337146880	12219793925519084442852695040	14153725415333992667331469824	12138112692460358840896598784	7803560005308187210029916800	3760865490604776669584133120	1341939474076621859202129920	344576455311298062796259328	60293776843891887663218688	6443697678987760238592000	317651255653438586880000		63568895624017156567955892096
14	58471088483926850159060598784	528833594319372116448163782656	2219919248902114923303394824192	5732607535544491826780511836160	10173848120752276978076398955520	13126703488435202782656492182016	12697608738670076482882032854784	9354437018570982932339087718528	5274145296047022844716781793280	2264661622548202729547034705920	728995806016390327794231672832	170587005043081162449769463808	27430498455729410744512413696	2713026079743077341200384000	124519292216147926056960000	62361086607160830593164730146176

Technical details

Node id: n-848ee0626a262d9f · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-95f9dd9edf6a7056

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:                1;
  1:                4,                 2;
  2:               52,                58,                16;
  3:             1456,              2560,              1492,               288;
  4:            71344,            172032,            155028,             61856,              9216;
  5:          5422144,          16641632,          20383728,          12452456,           3793216,            460800;
  6:        591013696,        2204332256,        3420023856,        2824946120,        1310037376,         323338752,          33177600;
  7:      87470027008,      384160516096,      722188091776,      753254363648,      470730251408,      176237798144,       36597482496,       3251404800;
  8:   16881715212544,    85339143063552,   188554847773056,   237818167931392,   187267497068688,    94267367222016,    29621752284160,    5311998885888,    416179814400;
  9: 4119138511860736, 23557588771938816, 59832324032921088, 88573518314494720, 84219812489645376, 53338572127299360, 22499021047301632, 6094851598190592, 962091694227456, 67421129932800;

%F Formula

The row polynomials satisfy P(n,x) = (3*n^2+1+2*n^2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3*n^2 + 1 + 2*n^2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3*n^2 + 1 + 2*n^2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - ↑ Geometric basis map (elementary) (above)
  - ↑ Geometric basis map (elementary) (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.